Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute
Original practice questions written to the published syllabus · not past papersThe optimisation theory statisticians use: linear programming with simplex and duality, transportation and assignment, integer and goal programming for farm and crop planning, non-linear optimisation with Kuhn-Tucker conditions, and the numerical optimisers inside likelihood estimation. The Agribusiness track teaches operations research for managers; this goes to the theory.
In the exams: AICE-JRF Ag Stat Unit 2 'linear programming: formulation, graphical, simplex, duality, transportation and assignment' (heavy); UPSC ISS Statistics III operations research (heavy, reported); ASRB NET Ag Stat (present)
How our questions for Indian Statistical Institute admission tests spread across the units of Optimization Techniques for Statisticians. Difficulty: 1 easy, 23 medium, 7 hard.
| Integer programming: branch and bound | 5 | |
| Dynamic programming | 5 | |
| Numerical optimisation: gradient, Newton and quasi-Newton methods | 5 | |
| Non-linear programming: Lagrange multipliers and Kuhn-Tucker conditions | 4 | |
| Goal programming for multi-objective farm planning | 3 | |
| Linear programming: formulation and graphical solution | 2 | |
| Simplex method, Big-M and two-phase methods | 2 | |
| Duality and sensitivity analysis | 2 | |
| Quadratic programming and risk programming (MOTAD) | 2 | |
| Transportation and assignment problems | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What shape is the feasible region of a linear programme?
C. A convex polyhedron
Intersections of half-spaces are convex polyhedra.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
Mustard's return stays at 30. Over what range of wheat return does the plan (4, 8) remain optimal?
A. 30 to 60
The plan stays optimal while c_wheat/c_mustard lies between the slopes 1 (land) and 2 (labour), i.e. 30 to 60.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
In the Hungarian method, when has an optimal assignment been reached?
D. When n lines cover the zeros
n covering lines guarantee n independent zeros for a complete assignment.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
When is a branch-and-bound node fathomed?
A. Infeasible, integral, or no better bound
Each of the three conditions shows that no better integer solution can lie in that branch.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
A binary-variable model requires that at most 2 of 5 irrigation projects be chosen. How is this written?
D. δ₁ + δ₂ + δ₃ + δ₄ + δ₅ ≤ 2
With δⱼ ∈ {0, 1}, the sum counts chosen projects.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
What does the Lagrange multiplier of a constraint measure?
B. The constraint's marginal value
λ = ∂f*/∂b, the shadow price of the constraint.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
A variance component must stay positive during optimisation. What is a standard device?
C. Optimise over its logarithm
Transforming constrained parameters (log σ², logit p) lets unconstrained optimisers work.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
After an MLE optimiser converges, why check that the Hessian is negative definite?
C. To confirm a true maximum
A stationary point may be a saddle or minimum; the inverse of −H gives the standard errors only at a proper maximum.
Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization
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